Kesirli Cique kollokasyon metodu ile uzay kesirli Schrodinger denkleminin sayısal çözümü

Öz

Bu çalışma, Uzay Kesirli Schrödinger Problemi'nin (UKSP), Kesirli Clique Kolokasyon Yöntemi (KCKM) ile sayısal çözümünün elde edilmesini konu almaktadır. Yöntem, ilk aşamada problemi, kesirli Clique polinomları temelinde kolokasyon noktaları kullanılarak bir adi diferansiyel ve cebirsel denklem sistemine dönüştürmekte; ikinci aşamada ise bu sistem Rezidüel Kuvvet Serileri Yöntemi (RKSY) ile çözülmektedir. Böylece KCKM, iki sayısal tekniğin birleşiminden oluşan hibrit bir yaklaşımdır. Yöntemin etkinliği ve üstünlüğü, sunulan nümerik örnekler üzerinden tartışılmıştır.

Anahtar Kelimeler

• Uzay kesirli Schrodinger denklemi
• Cilque polinomları
• Rezidüel kuvvet serisi yöntemi

Numerical Solution of Space Fractional Schrodinger equation by fractional Clique collocation method

Öz

This research presents the establishment of numerical solutions for space fractional Schrodinger problem (SFSP) by utilizing significant collocation methods, called fractional Clique collocation method (FCCM). First of all, the SFSP is reduced into a system of ordinary differential and algebraic equations by means of fractional Clique polynomials with collocation points. Secondly, the resulting system is solved numerically by Residual power series method (RPSM).Therefore, FCCM is a combination of fractional Clique polynomials and RPSM. Finally, illustrative examples are given to present how FCCM is striking and appealing.

Anahtar Kelimeler

• Space fractional Schrodinger equation
• Clique polynomials
• Residual power series method

Kaynakça

1. [1] Brunner, H., Iserles, A. & Norsett, S.P. (2011). The computation of the spectra of highly oscillatory Fredholm operators. Journal of Integral Equations and Applications, 23, 1-40.
2. [2] Wang, Y.X. & Fan, Q.B. (2012). The second kind Chebyshev wavelet method for solving fractional differential equations. Applied Mathematics and Computation, 218, 8592–8601.
3. [3] Yi, M.X. & Chen, Y.M. (2012). Haar wavelet operational matrix method for solving fractional partial differential equations. Computer Modeling in Engineering and Sciences, 88(3), 229–244.
4. [4] Momani, S. & Odibat, Z. (2007). Generalized differential transform method for solving a space and time-fractional diffusion-wave equation. Physics Letters A, 370, 379–387.
5. [5] Odibat, Z. & Momani, S. (2008). Generalized differential transform method: Application to differential equations of fractional order. Applied Mathematics and Computation, 197, 467–477.
6. [6] Odibat, Z. & Momani, S. (2008). A generalized differential transform method for linear partial differential equations of fractional order. Applied Mathematics Letters, 21, 194–199.
7. [7] Cetinkaya, S. & Demir, A. (2025). A new approach for the fractional Rosenau–Hyman problem by ARA transform. Mathematical Methods in the Applied Sciences.
8. [8] Cetinkaya, S. & Demir, A. (2023). On the solution of time fractional initial value problem by a new method with ARA transform. Journal of Intelligent and Fuzzy Systems, 44(2), 2693-2701.

9. [9] Cetinkaya, S. & Demir, A. (2023). On applications Shehu variational iteration method to time fractional initial boundary value problems. Konuralp Journal of Mathematics, 12(1), 13-20.
10. [10] Cetinkaya, S. & Demir, A. (2022). Effects of the ARA transform method for time fractional problems. Mathematica Moravica, 26(2), 73-84.
11. [11] Zhang, Y. (2009). A finite difference method for fractional partial differential equation. Applied Mathematics and Computation, 215, 524–529.
12. [12] Odibat, Z.M. (2010). A study on the convergence of variational iteration method. Mathematical and Computer Modeling, 51, 1181–1192.
13. [13] El-Sayed, A. (1998). Nonlinear functional differential equations of arbitrary orders. Nonlinear Analysis, 33(2), 181–186.
14. [14] El-Kalla, I.L. (2011). Error estimate of the series solution to a class of nonlinear fractional differential equations. Communications in Nonlinear Science and Numerical Simulation, 16, 1403–1408.
15. [15] Wazwaz, A.M. (2002). Partial Differential Equations: Methods and Applications. Balkema Publishers.
16. [16] Wazwaz, A.M. (2008). A study on linear and nonlinear Schrödinger equations by the variational iteration method. Chaos, Solitons and Fractals, 37(4), 1136-1142.
17. [17] Diudea, M.V., Gutman, I. & Lorentz, J. (1999). Molecular Topology.
18. [18] Harary, F. (1969). Graph Theory. Addison-Wesley.
19. [19] Podlubny, I. (1999). Fractional Differential Equations (Mathematics in Science and Engineering, Vol. 198). Elsevier.
20. [20] Caputo, M. (1969). Elasticita e Dissipazione. Zani-Chelli.
21. [21] Miller, K.S. & Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley.
22. [22] Kilbas, A.A., Srivastava, H.M. & Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations (Vol. 204). Elsevier.
23. [23] Mainardi, F. (2010). Fractional Calculus and Waves in Linear Viscoelasticity (1st ed.). Imperial College Press.